Front Mech Eng DOI 10.1007/s11465-017-0408-3 REVIEW ARTICLE Liangchi ZHANG, Weidong LIU Precision glass molding: Toward an optimal fabrication of optical lenses © The Author(s) 2016 This article is published with open access at link.springer.com and journal.hep.com.cn 2016 Abstract It is costly and time consuming to use machining processes, such as grinding, polishing and lapping, to produce optical glass lenses with complex features Precision glass molding (PGM) has thus been developed to realize an efficient manufacture of such optical components in a single step However, PGM faces various technical challenges For example, a PGM process must be carried out within the super-cooled region of optical glass above its glass transition temperature, in which the material has an unstable non-equilibrium structure Within a narrow window of allowable temperature variation, the glass viscosity can change from 105 to 1012 Pa$s due to the kinetic fragility of the super-cooled liquid This makes a PGM process sensitive to its molding temperature In addition, because of the structural relaxation in this temperature window, the atomic structure that governs the material properties is strongly dependent on time and thermal history Such complexity often leads to residual stresses and shape distortion in a lens molded, causing unexpected changes in density and refractive index This review will discuss some of the central issues in PGM processes and provide a method based on a manufacturing chain consideration from mold material selection, property and deformation characterization of optical glass to process optimization The realization of such optimization is a necessary step for the Industry 4.0 of PGM Keywords precision glass molding, optical lens, constitutive modeling, optimization, manufacturing chain, Industry 4.0 Received August 30, 2016; accepted October 10, 2016 ✉ Liangchi ZHANG ( ), Weidong LIU Laboratory for Precision and Nano Processing Technologies, School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney NSW 2052, Australia E-mail: Liangchi.zhang@unsw.edu.au Introduction Advanced glass lenses are important components to many modern technologies [1–5] A lithography process in the semiconductor industry, for example, relies on the quality of its optical system [6–9] Reducing the feature size of a microchip from 42 to 10 nm using the ultraviolet lithography technique requires that the key optical element of the lithography system needs a flatness of nm across an area of 30 cm [10] An advanced laser system demands high-quality microlens arrays to enhance the laser intensity [11–14] The development of space telescopes calls for ultra-precision, large-scale glass lens to explore the universe [15–18] Electronic devices in the consumer industry are also examples of application of precision lenses, such as the aspherical lenses for smart mobile phones and large-scale liquid crystal display panels [1,19– 21] Traditional methods of fabricating glass lenses are grinding, polishing and lapping [22–25], dating back to the 1500s when microscopes [26,27] and telescopes [28– 30] were invented However, lens polishing skills were among talented craftsmen [26,28], developed from their life-long experience The invention and application of grinding and polishing machines in the 1800s largely improved the efficiency of lens production [26,28]; these machines at that stage could only produce spherical lenses with rough surfaces To avoid spherical aberration, one had to use a set of spherical lenses in an optical device [1] The recent advances in computer numerical control machining have overcome many of the difficulties in the production of precision glass lenses [31,32] However, the processes of precision machining, including single-point diamond turning, grinding, polishing and lapping, is still very time consuming and expensive An aspheric optical element can easily cost some thousands of dollars [33] Moreover, single point diamond turning, which is flexible to make precise and complex micro/nano features on a lens surface, is not suitable for the most needed optical material, the Sibased optical glass This is because silicate optical glass Front Mech Eng can easily produce severe cleavages and microchipping on a diamond tool surface and therefore bring about tool wear [34–36] The technique of precision glass molding (PGM) has been developed for the manufacture of aspherical and irregular glass optics [37–39] This technique is based on the softening of glass in its super-cooled liquid region which is above the glass transition temperature Tg [40] With this, once the surface of a mold cavity is made to have the required dimensions, geometries and features, in a single production step a precision glass lens can be thermally formed to copy the cavity geometries and features of the mold Figure is a comparison of the PGM with the traditional machining process It is clear that the PGM can significantly reduce the production time and cost chain, as shown in Fig 2, i.e., mold material selection (Section 3), property and deformation characterization of optical glass (Section 4), and process optimization (Section 5) Fig 2 Fig A comparison of traditional machining process with PGM It has been found, however, that the quality of a molded lens depends on many factors in its PGM process, such as the mold material selection, the mold quality, the glass property change in PGM and the process control including temperature and pressure variations [41] Any imperfection in each will influence the quality of a lens molded A PGM process involves heating, soaking, molding, first cooling, demolding and final cooling, through which both the glass and mold experience complicated thermo-mechanical deformation For example, in a very narrow temperature range the viscosity of glass undergoes a change of several orders of its magnitude, which brings about significant challenges for the production of high precision lenses by PGM [40] As a result, the quality of the molded lenses is still not as good as those manufactured by precision machining processes, although the production cost of the former is only half of that of the latter [42] It has therefore been a major research effort in the optics manufacturing field to try to greatly improve the PGM technology for making optics of the same or even higher quality compared with the products from the ultra-precision machining approach The brief discussion above shows that to produce precision lens by PGM, an optimal process based on a comprehensive consideration of the PGM manufacturing chain is necessary Thus, after an introduction to the PGM in Section 2, this paper will focus on the investigations into the central problems individually in the manufacturing Key factors in the manufacturing chain of a PGM process Precision glass molding PGM is a thermal forming process, which involves the heating of an optical glass preform to above its glass transition temperature (Tg), the compression forming of the preform in a mold cavity mechanically, and the cooling and demolding of the formed element A PGM process needs to be carried out in a precisely controlled environment [41], including thermal and mechanical loading-unloading This section will briefly introduce some commonly used moldable optical glasses, the typical procedure of PGM process and the achieved performance of molded lenses 2.1 Moldable optical glass and preforms Most optical glass materials have super-cooled liquid regions, in which the materials become soft and moldable [33,41] In production, the efficiency and cost are important factors to be considered In general, optical glass with a lower Tg has a higher moldability, with which the molding temperature is lower, the shape distortion due to cooling is smaller, the material property change in PGM is less, and the mold service life is longer Table [41] lists the definition of moldable glass supplied by a number of glass manufacturers Figure [33] summarizes the moldable glasses available with their key optical properties, index of refraction nd and Abbe number vd Table Moldable glass defined by manufacturers [41] Manufacturer Prefix Definition D Tg < 618 °C Hikari Q Tg < 607 °C Hoya M Tg < 560 °C Ohara L Tg < 608 °C Schott P Tg < 550 °C Sumita K Tg < 530 °C CDGM Liangchi ZHANG et al Process optimization of precision lens molding After the selection of a glass material, a proper glass preform for a specific optical lens/component needs to be determined, of which the quality will have a direct influence on that of the product There are many different preforms in terms of shapes and sizes, such as a ballpreform The advantages of a ball-preform are (1) that its spherical shape can be easily deformed in PGM into many commonly used lens geometries; and (2) that the ballpreform manufacture is already mature — High quality and low cost Most suppliers can provide ball preforms of diameters from to mm [41] However, very small or large preforms are still difficult to manufacture [41] Flat preforms of optical glass are also commonly used in PGM Compared to the ball-preform, a flat preform can be polished to a very high surface finish, and thus can be used for molding micro/nano micro-lens arrays, V-groove arrays, and other thin components [41] Flat preforms are also suitable for molding diverging lenses because a flat surface can be deformed easily to either a concave or a convex shape [41] To mold an optical component with a complex geometry or of a large dimension, a near-net shape preform is often required to minimize the geometrical change in PGM, although producing a near-net shape preform is expensive 2.2 Typical PGM procedure The proper glass preform now allows to have the PGM to happen on a molding machine A PGM is a hightemperature compression forming process in a controlled environment [41], which includes heating, soaking, molding, cooling, demolding and final cooling, as shown in Figs and The chamber of the PGM machine is evacuated with an inert gas, such as nitrogen, before heating the preform and mold up to the required molding temperature above the glass Tg This will make the glass viscosity be in the range of 107 to 108 Pa·s [40] The process is then conducted by applying either a constant compression force or a pressing velocity within a period of time selected At the end of the molding, the lens is first Fig Moldable optical glass available [33] Fig A typical processing cycle of PGM cooled down with a small rate, during which the compressive force remains until the glass temperature has dropped to a specific temperature corresponding to the strain point of glass (h = 1013.5 Pa·s) The demolding process can then take place associated with a higher cooling rate for a higher production efficiency [40,41] 2.3 Performance of molded optical lenses Most of the standard lenses can be manufactured by PGM with a reasonable quality, including biconvex, planoconvex, plano-concave or meniscus It is cost-effective if a lens can be molded by using a preform commercially available The manufacture of a mold cavity with very small or steep features is difficult Large lenses also add difficulties and production cost due to the increased difficulties in quality control The maximum size of molded lens was in the order of 100 mm in diameter [42] Molding diffractive lenses is still challenging because grinding complicated diffractive features in the cavity surface of a carbide or ceramic mold is very difficult [43] Table [41] shows some typical tolerances of the lens by PGM It should be noted that the tolerances depend on many factors and those in the table are not exhaustive For example, the size and shape of a lens can have a significant impact on the tolerance actually achievable In PGM, mold deterioration and shape and optical property variations of molded lenses are major problems [33,40,44–47] The manufacture of a quality optical mold is the costliest part in the PGM production chain For example, a general PGM mold that costs about 4000 USD can fail within 1000 of molding cycles [48] Thus, the selection of a suitable mold material is central to both the production quality and cost The geometry accuracy of a lens molded is related directly to the complicated deformation of the glass material during the molding and cooling in a PGM, which cannot be clarified by a trial-anderror approach Moreover, molding introduces residual stresses in a lens molded, which in turn alters the refractive Front Mech Eng Fig A typical PGM process Table Typical tolerances for a precision glass molded lens [41] Center thickness /mm Diameter /mm Decentration /mm Wedge /(′ ) Power/irregularity /fringes Surface roughness/nm Surface quality (scratch$dig–1) Commercial Ỉ0.050 Æ0.025 Æ0.020 Æ10 5/2 20 60–40 Precision Æ0.025 Æ0.015 Æ0.010 Æ3 3/1 10 40–20 index and Abbe number of optical glass [41,45,46,49–51] Hence, a sophisticated study on the complicated mechanical behavior of optical glass in PGM is imperative These problems will be addressed individually in the following sections Mold material selection A mold replicates the optical prescription of a lens thermally formed and thus directly affects the optical quality of the product The basic requirements for a mold of PGM are: (1) Excellent mechanical properties and wear resistance, (2) small coefficient of thermal expansion, (3) outstanding thermal stability, and (4) good machinability 3.1 Mold materials The first consideration in selecting a mold material is the Tg of the glass to be molded As shown in Table [41], it can be an ultra-low-Tg PGM, a low-Tg PGM or a high-Tg PGM Electroless nickel-phosphor is a commonly used mold material for an ultra-low-Tg PGM, because this material is hard, and has high wear and corrosion resistance at a moderate temperature [52–54] It has also a high machinability for single point turning [55,56] However, this material can be crystallized and annealed at a temperature above 400 °C; after which its properties are deteriorated [57] Thus, electroless nickel-phosphor is suitable only to ultra-low-Tg PGM processes Compared with the mold material property requirement by an ultra-low-Tg PGM, the low-Tg PGM calls for a much harder substrate Hence, tungsten carbide (WC) [58] and ceramics such as silicon carbide (SiC) [59,60] are often used However, because of the hardness and brittleness of these materials, their machinability by traditional methods is poor In general, using a pre-shaped green mold can reduce significant workload in achieving the optical surface finish by precision grinding [41] Depending on the requirements for molding specific lenses, polishing may need to be used to remove any residual grinding defects and minimize surface roughness Generally, optical glass with a high Tg of greater than 620 °C is not suitable for PGM because of the material’s low moldability but high cost Fortunately, lower Tg glass can be used in most cases [41] apart from some specific applications which require the use of high Tg glass (e.g., quartz glass with Tg = 1200 °C) Some people suggested that amorphous glass/carbon molds can work at a high temperature up to 1500 °C [61] However, a recent investigation found that oxidation-induced property deterioration and thermal mismatch problems of this types of materials will cause instability of the molds [62] At a high temperature, a mold can wear quickly in its direct interactions with glass workpieces during molding Table Selection of mold materials for different PGM processes [41] Process Tg of glass Molds Ultra-low Tg Low Tg High Tg < 400 °C 400 °C < Tg < 620 °C Tg > 620 °C Electroless nickel-phosphor Carbides or ceramics Carbides or ceramics Single point diamond turning Micro-grinding Micro-grinding Cost Low High Very high Tooling life Low Medium Very low Manufacturing process Liangchi ZHANG et al Process optimization of precision lens molding Thus, coating is often used for extending mold life and thereby leading to the cost reduction of the molded optical products There are mainly three types of coatings available in the market, namely, ceramic coating (TiAlN, CrN, TiBCN or TiBC) [63], noble metal coating (Pt/Ir [64,65] and Re/Ir [66,67]), and carbon coating (diamondlike coating, amorphous carbon coating) [68] Ceramic coatings have been widely used in making machining tools, and therefore are straightforward to apply to PGM molds However, most ceramic coatings are prone to adhesion with glass [64] Noble metal coatings, especially the Pt/Ir alloy coating, can avoid such adhesion [64,65] Nevertheless, it has been found that the Pt/Ir layer is not stable [66,67], and that the substrate material can diffuse into the coating layer and reduce its quality However, the noble metal coating of Re/Ir demonstrates very stable properties [66] and low wetting angle among various coating materials [67] Diamonds-like coatings require a vacuum environment to avoid oxidation at high temperature Its performance still needs further studies [68,69] 3.2 Selection of mold materials The wear and fatigue of a mold are due to the cyclic mechanical stressing, and harsh heating and cooling in PGM production An appropriate assessment or test is thus important when selecting proper mold/coating materials The cyclic mechanical loading-unloading on a mold surface is through the mold-workpiece mechanical interactions (pressure and friction), which can result in the deformation, surface wear and fatigue failure of the mold The repeated heating-cooling cycles, in thousands normally, not only bring about significant thermal stress variations in the mold, but also promote chemical reaction (e.g., diffusion and corrosion [48]) and adhesion between the mold and workpiece The chemical reactions are mainly adhesion of glass on the coating, processes in the coating, and corrosive attack at the coating surface These, in turn, can largely shorten the mold life In addition to the wear and fatigue considerations, the coefficient of thermal expansion (CTE) of the mold material, including that of the coating, must also be taken into account carefully Ideally, the whole molding system composed of mold, coating and optical glass has the same CTE so that the glass-mold friction can be minimized and the accuracy of the molded optical lens can be maximized This of course is impossible to achieve in practice, but aim to minimize the mismatch of the CTEs Some studies have been carried out to understand the effects of the above on the service life of PGM molds These include the investigations on the sticking behavior of mold materials/coatings with hot glass preforms by the frequent contact method [70], the performance of different coatings in PGM [66,71], and the anti-sticking ability of Pt/Ir and TiAlN coatings on tungsten carbide and silicon wafer substrates [65] It was understood that compression hold time, cooling time and peak force can significantly affect the sticking However, most of the above works were conducted at a low temperature in a non-isothermal environment and as such the results are not directly applicable to a real PGM process Recently, a quick testing facility [72,73] was proposed to assess the service life of mold coatings for PGM It was noticed that in PGM a long period of time in the heating-cooling process is without mechanical stresses or chemical influence From the pure mechanical wear and fatigue point of view, therefore, a cheap and simple testing facility, involving mechanical loading and unloading strokes only, may be used to bypass the time-consuming stages of heating and cooling The performance of three standard coatings (TiAlN, CrAlN, and Pt/Ir) on flat WC pins was studied by using this type of testing, for the molding of B270 glass Figure [73] presents the images of the mold (pins) and glass imprints after 20 pressing steps, which shows clearly that the WC pins with Nitrogen coatings were severely worn Edge damages and partial imprints took place on the corresponding glass specimen surfaces However, the WC pin with the Pt/Ir coating remained undamaged Property and deformation of optical glass With a proper mold selected, the second critical part in the manufacturing chain of optical lenses, as emphasized in Fig 2, is to achieve an accurate understanding of the variation mechanisms of property and deformation behavior of optical glass in a PGM process Otherwise, many critical issues which influence greatly the residual stresses, geometry distortion and optical properties of a molded lens cannot be controlled [33,40,44–47] Many studies and production process designs have been trial-and-error, highly dependent on the practical experience and skills For example, it usually needs to months of laborintensive refining process to reach a satisfactory mold geometry to compensate the shape deviation of a molded optical lens [74], at the cost of about 4000 USD [74] It has been recognized that computer simulation can minimize the trial-and-error design process [40,75,76] To obtain sophisticated solutions and useful guidelines for process optimization, the finite element (FE) method has been widely used to reveal the mechanism of geometry deviation and residual stresses [77,78] This involves an accurate constitutive description of optical glass, the instantaneous property change of optical glass during the heating-cooling cycle in PGM, and lens distortion characterization These will be discussed in the following sections 4.1 Constitutive modeling of optical glass To make a reliable numerical simulation, it is essential to Front Mech Eng Fig Performance comparison of TiAlN, CrAlN and Pt/Ir coatings after 20 pressing steps [73] use reliable constitutive models that can accurately describe the behavior of optical glass throughout a PGM process However, to establish such models is challenging because the behavior of optical glass during the thermalmechanical deformation in PGM is strongly nonlinear and complicated In general, a complete constitutive model of glass suitable for PGM should be able to the following relationships of mechanics quantities: (1) The thermoviscoelastic relationship of stress, strain, strain rate and temperature, and (2) the nonlinear temperature dependence of the material properties [40] A significant effort has been placed to develop constitutive models for describing the thermomechanical behavior of optical glass in PGM Some used measured thermo-viscoelastic properties of the materials (BK-7 and TaF-3 [75]), obtained the viscoelastic property of glass by using the relaxation data from a cylinder compression test with the assumption of incompressibility [76], or treated glass as an elasto-viscoplastic material to account for the strain rate effect [79] In most of these works the temperature-dependent rheology was modeled by the classical phenomenological Vogel-Fulcher-Tammann equation [37] or the thermos-rheological simple assumption [75,77], in which the parameters need to be obtained by curve fittings to a series of viscosity tests Recently, a method was proposed for identifying the shear relaxation modulus and the structural relaxation function via measuring the time variation of the glass plate thickness [80] The CTE variation was often modeled by the ToolNarayanaswamy-Moynihan (TNM) model [77,78], the parameterization of which needs structure relaxation tests and thermal expansion tests It is clearly complicated to establish a constitutive model using these methods A modulus-based constitutive model, as summarized in Table 4, was recently developed for analyzing PGM processes numerically [40] The core of this approach is that all the temperature-dependent material properties are determined by the relationship between the elastic moduli and microstructure of a material In this model [40], the strain tensor and stress tensor are divided into volumetric and deviatoric parts The relationship between deviatoric stress and strain is described by a standard linear solid (SLS) model [81] Because of the strong resistance to volumetric changes, the bulk viscosity of optical glass (PBK7) can be considered to be infinite, and thus a simple thermal elastic relationship is enough Temperaturedependent Young’s modulus, shear modulus, and bulk modulus can be measured straightforwardly by an impulse Table Modulus-based constitutive model for optical glass [40] Relationship Stress and strain Equation εij ¼ eij ỵ trịij =3, ij ẳ Sij ỵ trịij =3 Viscosity variation trị=3 T ẳ trị=9K Sij Sij G G ỵ r e_ ij ỵ r eij ẳ ỵ G s 2G 2s s ẳ expVc G1 T ị=kB T ị Thermal expansion ẳ G þ ðαL – αG ÞδTf =δT Volumetric relationship Deviatoric relationship Structure relaxation description À Á dT ðT Þ Tf ¼ T – !ðT Þ Mp – 0 d d0 t ¼ ! 1=τ p dt# Mp ị ẳ expẵ = pr ị p ẳ expẵxH=RT ỵ xịH=RTf εij – Strain tensor; sij – Stress tensor; eij – Deviatoric strain; Sij – Deviatoric stress; tr(ε) – The trace of the strain tensor; tr(s) – The trace of the stress tensor; dij – Kronecker delta; K – Bulk modulus; α – The coefficient of thermal expansion; T – Temperature; Gr – The modulus in the elastic branch of the SLS model; G – The shear modulus in the Maxwell branch of the SLS model; hs – Shear viscosity; h0 – Reference viscosity; kB – Boltzmann constant; Vc – Characteristic temperature-independent microscopic volume; G1(T) – Instantaneous shear modulus; αG – The reference CTE at low temperature glassy state; αL – The reference CTE at high temperature liquid state; Tf – Effective temperature; T0 – The reference temperature; Mp(x) – The structural relaxation function; x – Reduced time; – Structural relaxation time; ΔH – The active energy; R – The ideal gas constant; t0, x, b – Constants Liangchi ZHANG et al Process optimization of precision lens molding excitation method [82] Based on the shoving model [83], the temperature-dependent viscosity of optical glass can then be directly linked to its shear modulus, and the CTE of glass can be predicted through modulus based on a phenomenological TNM model [77,78], in which the parameters needed in TNM model can be determined by the modulus changes along with the temperature in the impulse excitation method The above constitutive model with the measured/derived parameters has been verified and programmed into ABAQUS as a user material (UMAT) [40] 4.2 Mechanisms of lens distortion Lens accuracy, including lens geometrical accuracy and quality of its surface finish, is critical Ultra-precision grinding, polishing and lapping can achieve high lens accuracy step by step at a high cost The surface and shape accuracy of a lens by PGM, however, are formed in a single thermal forming step at high temperature [40] In general, the quality of a molded lens, both surface and shape accuracy, depends largely on that of the mold surface and the lens distortion during annealing, cooling and demolding It was reported that the shape derivation of a molded lens can be as high as 20 mm, about 20 times higher than the deviation allowed according to the optical design specifications [74] Thus, in designing a mold, the cavity cannot be simply the dimensions of the required geometry, but must include a compensation taking into account the distortion of the lens in PGM Such compensation at the initial model design can be realized with the aid of numerical simulation using a proper constitutive model For example, the formation mechanism of shape deviation of lenses in PGM was investigated in detail by using the modulus-based constitutive model [40] Figure 7(a) shows the evolution of shape variation of a lens during a typical PGM process In the pressing (molding) stage, the glass ball (preform) was compressed to comply with the mold cavity The subsequent demolding did not lead to a significant shape deviation However, in the cooling stage, a large shape deviation occurred near the center of the lens, as shown in the figure insert When the internal temperature of the lens reduces to below the material’s Tg, no further deviation occurs The deviation details with respect to the mold in the radial direction are presented in Fig 7(b) The large deviations near the center and the edge of the lens are due to the cooling-induced shrinkage and edge effect, respectively [40] For a precision lens, the allowed center thickness change is about 25 mm [41], and the maximum deviation of overall surface shape should be within several micrometers or smaller [77,78] Thus, the above numerical analysis [40] has demonstrated that mold compensation is essential; otherwise the lenses by PGM are not usable The numerical analysis [40] can also identify the geometry effect and key processing parameters that Fig (a) The evolution of a lens shape in PGM; (b) the deviation with respect to the mold cavity geometry; (c) the relationship between H/R and r/R [40] influence the final shape As shown in Fig 7(c), the relationship between H/R and r/R is almost linear, in which H is the final thickness of the molded lens, R is the curvature radius of the mold, and r is the radius of the glass ball-preform This dimensionless result shows the geometry similarity of the glass molding process, indicating that if one gets the final shape of a lens at a certain dimension, lenses of the similar geometry of other dimensions can be predicted by this linear relationship if their forming conditions are the same It should be noted, however, that this linear relationship is the result of a macroscopic analysis, and that nonlinear effects may arise in micro-lens It has also been reported [77,78], based on a parametric study on the formation mechanism of shape deviation, that Front Mech Eng the structural relaxation of glass is the primary reason for lens distortion in PGM [78] The activation energy constant and relaxation time constant in the TNM model are key parameters of structural relaxation that affect the lens shape change [78,82] Hence, glasses with different values of these parameters must be compensated differently It is thus essential to have the structural relaxation parameters well defined to predict the deviation within tolerance A novel method [82] has been developed recently to identify these parameters based on an impulse excitation technique Studies [77,78] have suggested that the most critical stage to introduce lens distortion is at the beginning of demolding The thermal expansion coefficients of the mold material and internal stresses of the lens play an important role in the shape deviation of lens Other important factors include molding temperature, loading-unloading paths and cooling rates 4.3 Internal property change As mentioned in Section 2, some critical optical properties of glass can be changed after molding This is because the cooling rate of glass material in PGM is different from that of glass preform Most glass preforms have been well annealed by manufacturers However, in PGM, fast cooling rates are often used to increase production efficiency and reduce cost During the cooling stage of PGM, glass properties such as the CTE can change due to Fig structural relaxation and lead to internal residual stresses [84–86] It has been reported that residual stresses can severely alter the local density, and lead to inhomogeneous refractive index in an optical lens [84] For instance, a residual stress of MPa in P-BK7 glass lens can bring about a variation of refractive index of  10–4, and thus produce unwanted changes in the light path, intensity, and deterioration of image quality [87,88] Therefore, it is important to understand the formation mechanism of residuals stresses in PGM process and its effect on optical properties Some studies showed that the duration of cooling from the molding temperature to Tg is important in minimizing residual stresses [75] and that the residual stresses in a molded lens can be controlled to a very small value if a proper cooling is applied [84] Further, the evolving internal stresses within glass can be affected by changing the rheology behavior of glass at molding temperature, the friction at the glass/mold interface, and the time/temperature at which the demolding is applied [77,78] A recently comprehensive investigation [40] revealed the formation mechanisms of residual stresses as well as some key parameters that affect the residual stresses Figure [40] shows some typical distributions of residual hydrostatic and von Mises stresses in a convex-convex lens by PGM It can be seen that the inner part of the lens sustains tensile residual stresses, but its external surfaces are under high compressive stresses (Fig 8(a)) The region between these two has low residual stresses, which is also The distributions of residual (a) hydrostatic stress, and (b) von Mises stress [40] Liangchi ZHANG et al Process optimization of precision lens molding true in the von Mises stress distribution (Fig 8(b)) [40] The two minima of the von Mises stresses locate closely to the top and bottom subsurfaces, symmetrically It should be noted that lenses for different shapes can have very different distributions of residual stresses The formation mechanism of the residual stresses can be understood by monitoring the evolutions of the von Mises stresses in the lens For convenience, let us investigate the stresses at the top, middle and bottom points of the lens [40] as shown in Fig 9(a) It can be seen that the internal stresses before 270 s are very small except in the initial pressing stage At around 270 s (in cooling stage), however, the internal stresses increase to a plateau till the end of the PGM process to form residual stresses Figure 9(b) presents the internal stress distributions along the central line through the lens thickness at three different times around 270 s It is clear that both the magnitude and gradient of the internal stresses increase significantly in this region The sharp internal stress increase is closely related to the heterogeneous evolution of CTE of the optical glass during PGM [40] As shown in Fig 9(c), in the time interval between 270 and 320 s, the CTE decreases quickly when the lens temperature approaches Tg in the cooling stage As the temperature distribution in the lens is inhomogeneous during glass molding, the changes of the CTE at different positions are asynchronous The difference of CTEs reaches the maximum at 280 s as shown in the insert of Fig 9(c), corresponding to the significant increase of the magnitude and gradient of the internal stresses Since residual stresses arise due to the sharp increase of internal stresses during cooling, it is reasonable to expect that residual stresses can be reduced by controlling the cooling rate It has been found that the rate of the first cooling stage from the molding temperature to Tg is very important in minimizing the residual stress [75] If this cooling stage can be of a sufficient duration, the second cooling stage from Tg to room temperature can be shorter [75] A recent study [40] has explicitly shown the different effects of the two cooling stages on the formation of residual stresses (Fig 10) Figure 10(a) demonstrates the evolutions of the von Mises stresses in a lens under three different cooling rates in the first cooling stage, but with a constant cooling rate of °C/s in the second cooling stage It is clear that the internal stresses and residual stresses decrease if the first stage cooling rate is smaller (above Tg) Figure 10(b) shows that varying the second stage cooling rate (below Tg) has a negligible effect on the internal and residual stresses Thus, to effectively minimize the residual stresses in a lens, a good strategy would be to use a small cooling rate in the first stage, and then a larger cooling rate in the second stage for the sake of production efficiency [40] Fig (a) Variations of the von Mises stresses with time at different points in the lens; (b) the stress distributions along the central line through the lens thickness; (c) the variations of CTEs at different points with time [40] Process optimization The quality of a lens manufactured by PGM is influenced by a series of factors such as the quality of the optical glass preform, quality of the mold (design, material and fabrication) and processing conditions/parameters of the molding process Although the mechanisms of mold deterioration, lens shape distortion and residual stress have been studied, it is essential to make full use of the mechanisms explored in lens production to compensate the possible quality deviation of a lens from the beginning of a 10 Front Mech Eng Fig 11 A typical optimization process selecting appropriate PGM parameters These will be discussed individually below 5.1.1 Fig 10 The effect of cooling rate on the internal stresses with time: (a) Effect of cooling rates in the first cooling stage, and (b) effect of cooling rates in the second cooling stage [40] PGM process design As have been discussed in detail in the previous sections, the relationships between the product quality and control factors are complex and highly nonlinear Any trial-and-error approaches of compensation cannot work effectively A process optimization with the aid of a reliable numerical simulation is a cost-effective way to minimize the problems throughout the whole manufacturing chain of lens production In the following, we will use a simple example to demonstrate the PGM optimization process, with a single optimization objective, from the point of view of the manufacturing chain consideration 5.1 Optimization strategy A process optimization usually consists of three parts: Determining realistic objective functions, selecting reliable optimization algorithms, and defining key criteria for optimization As shown in Fig 11, to optimize a PGM process of lens, the objective functions are not simple equations [44,89,90] The criteria must be determined based on the objectives to be optimized Briefly speaking, it gets the parameters to be optimized from the optimization algorithms based on the criteria established to come up with a set of results required for the design of the mold and PGM processing parameters Generally, it is easier to optimize relevant factors for best values of a single objective, such as reducing the shape deviation by mold compensation or minimizing the residual stresses by Mold shape optimization According to the mechanism investigation highlighted previously, the lens shape distortion in PGM starts at the cooling stage due to the inevitable thermal shrinkage of optical glass Thus, the mold geometry and dimension must be optimized to compensate such effects [44,89,90] Different algorithms have been used for optimizing the mold shape, using, e.g., an iterative algorithm [91,92], a sequential quadratic programming method [90] or an iterative deviation method [44] An authors’ recent work (unpublished data) was completed by using a numerical optimization platform based on the simplex method and finite element simulation to give rise to the optimal design for producing a formulated aspherical lens surface defined by Eq (1), Y ðX ị ẳ X2 r! ỵ aX , X2 R ỵ ỵ k ị R (1) where X is the distance from the lens axis, Y is the Ycomponent of the distance from the vertex, R is the radius of curvature, k is conic constant, and a is the correction coefficient of high order terms The advantages of using the formulated aspherical mold shape are: (i) The number of the optimization parameters is much less than that in the optimization of node positions in a finite element simulation, and (ii) the optimized parameters can be directly used by an ultra-precision machining system for making a mold The profile-meansquare-deviation (PMSD), Eq (2), is selected as the optimization objective s Niẳ1 yi ^y i ị2 PMSD ¼